A Newly-Derived Cosmological Redshift Formula Which Solves the Hubble Tension and Yet Maintains Consistency with T_t = T_0(1 + z), the R_h = ct Principle and the Stefan-Boltzmann Law

Espen Haug; Eugene Tattum

doi:10.20944/preprints202412.0046.v1

Submitted:

01 December 2024

Posted:

02 December 2024

You are already at the latest version

Abstract

Numerous cosmological redshift formulae have been suggested in the field of cosmology. One of these is the well-known cosmological redshift formula used in the Λ-CDM model and in some R_h = ct models. In a recent type of “black hole” cosmology model, the redshift formula is derived from three fundamental principles: that the Stefan-Boltzmann law holds with respect to perfect and almost perfect black bodies; that the universe follows the Rh = ct principle; and that the time-dependent CMB temperature in relation to cosmological redshift is given by the observed relation T_t = T_0(1 + z). These three principles have recently been used by Haug and Tatum [1] to derive z = Sqrt(R_h/R_t) − 1, which has led to a simple but powerful solution to the Hubble tension [2–4]. In this short note, we build on our recent work and describe multiple different ways in which to represent the same cosmological redshift formula.

Keywords:

Cosmological redshif

;

Stefan Boltzmann law

;

R_h=ct cosmology

;

CMB temperature and z

Subject:

Physical Sciences - Astronomy and Astrophysics

Numerous cosmological redshift formulae have been suggested in the field of cosmology. One of these is the well-known cosmological redshift formula used in the

Λ

-CDM model and in some

R_{h} = c t

models. In a recent type of “black hole” cosmology model, the redshift formula is derived from three fundamental principles: that the Stefan-Boltzmann law holds with respect to perfect and almost perfect black bodies; that the universe follows the

R_{h} = c t

principle; and that the time-dependent CMB temperature in relation to cosmological redshift is given by the observed relation

T_{t} = T_{0} (1 + z)

. These three principles have recently been used by Haug and Tatum [1] to derive

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

, which has led to a simple but powerful solution to the Hubble tension [2,3,4]. In this short note, we build on our recent work and describe multiple different ways in which to represent the same cosmological redshift formula.

First, we will briefly explain how this redshift formula was derived. The following two formulae are based on the 2015 publication where by Tatum et al. [5] suggested:

\begin{matrix} T_{0} = \frac{ℏ c}{k_{b} 4 π \sqrt{R_{h} 2 l_{p}}} \end{matrix}

(1)

If we assume that the

R_{h} = c t

cosmological principle holds true, we must have:

\begin{matrix} T_{t} = \frac{ℏ c}{k_{b} 4 π \sqrt{R_{t} 2 l_{p}}} \end{matrix}

(2)

Subsequently, Haug and Wojnow [6,7] proved that the same formulae could be derived from the Stefan-Boltzmann law. To understand the significance of this, one must first be aware that the cosmic microwave background (CMB) radiation spectrum is that of a nearly perfect, black body, as pointed out by Muller et al. [8], who stated:

“Observations with the COBE satellite have demonstrated that the CMB corresponds a nearly perfect black body characterized by a temperature $T_{0}$ at $z = 0$ , which is measured with very high accuracy, $T_{0} = 2.72548 \pm 0.00057 K$ ."

This fact is also implied by the recent work of Dhal and Paul [9]). In addition to the Stefan-Boltzmann law and the

R_{h} = c t

principle, we have the observed relation between the current observed CMB temperature and the CMB temperature at a given distance, as a function of the observed redshift:

T_{0} = T_{t} (1 + z)

(3)

As pointed out by Riechers et al. [10]:

“ No deviations from the expected $(1 + z)$ scaling behaviour of the microwave background temperature have been seen, but the measurements have not extended deeply into the matter-dominated era of the Universe at redshifts $z > 3.3$ ."

Now, if we replace

T_{0}

and

T_{t}

with the Stefan-Boltzmann-derived formulae for these, we obtain:

\frac{ℏ c}{k_{b} 4 π \sqrt{R_{h} 2 l_{p}}} = \frac{ℏ c}{k_{b} 4 π \sqrt{R_{t} 2 l_{p}}} (1 + z)

(4)

That, when solved for z, gives:

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

(5)

This is the Haug and Tatum cosmological redshift equation, fully derived based on three important principles:

The Stefan-Boltzmann law must hold for nearly perfect black bodies. As we have noted, the cosmic microwave background is considered the most nearly perfect black body radiation spectrum observed, so this principle must apply.
The observed relation $T_{t} = T_{0} (1 + z)$ must hold. All evidence points in this direction, even though, based on observations alone, one cannot entirely rule out the possibility of $T_{t} = T_{0} {(1 + z)}^{1 - β}$ with $β$ slightly different from zero, see [10,11]. However, the probability of $β$ being slightly different from zero is very low, as this would lead to a significantly more complicated model and potential inconsistencies. At present, it appears to be highly probable that the relationship $T_{t} = T_{0} (1 + z)$ is correct and that any serious cosmological model must be consistent with this.
The final principle required to derive our cosmological redshift equation is our assumption of $R_{h} = c t$ cosmology. A series of observational studies appears to favor $R_{h} = c t$ cosmology. However, caution is needed here, since the Melia-type $R_{h} = c t$ cosmology differs in several significant ways from the Haug-Tatum $R_{h} = c t$ cosmology (HTC). One major difference is the redshift formula derived above and the fact that HTC can accurately predict the current CMB temperature.

If the

R_{h} = c t

principle holds true (and much points in that direction), then the evidence in favor of the

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1

cosmological redshift formula is strong. One can potentially solve the Hubble tension using the assumed

z = \frac{R_{h}}{R_{t}} - 1

or even

z = {(\frac{R_{h}}{R_{t}})}^{x} - 1

(see [4]), but the problem with such models then becomes that they are inconsistent with

T_{t} = T_{0} (1 + z)

for any x different than

\frac{1}{2}

. Or one can still potentially make it consistent with

T_{t} = T_{0} (1 + z)

for

x \neq \frac{1}{2}

, but then one cannot make such a model consistent with the observed and Stefan-Boltzmann law-derived current CMB temperature.

It is also worth noting that our cosmological redshift formula can be trivially rewritten in multiple forms:

z = \sqrt{\frac{R_{h}}{R_{t}}} - 1 = \sqrt{\frac{H_{t}}{H_{0}}} - 1 = \sqrt{\frac{t_{h, 0}}{t_{h, t}}} - 1 = \sqrt{\frac{M_{c}}{M_{c, t}}} - 1 = \sqrt{\frac{ρ_{c}}{ρ_{c, t}}} - 1 = \frac{T_{t}}{T_{0}} - 1

(6)

where

t_{h} = \frac{1}{H_{0}}

,

t_{t} = \frac{1}{H_{t}}

,

M_{c} = \frac{c^{2} R_{h}}{2 G}

,

M_{c, t} = \frac{c^{2} R_{t}}{2 G}

,

ρ_{c} = \frac{3 c^{4}}{R_{h}^{2} 8 π G}

and

ρ_{c, t} = \frac{3 c^{4}}{R_{t}^{2} 8 π G}

(critical energy density). Note that the cosmological redshift term z, as function of CMB temperature, is not squared. The reason for this is that the only variable in

T_{t}

is already related to

\sqrt{R_{t}}

. This relationship comes directly from the Stefan-Boltzmann law.

The equalities in Equation (6) are a direct result of our assumption of

R_{h} = c t

cosmology (a linear model), meaning that the percentage change in

R_{t}

,

H_{t}

,

t_{t}

,

M_{c, t}

, and

ρ_{c, t}

is the same as the percentage change in

R_{t}

.

In conclusion, the Haug and Tatum cosmological redshift is the only cosmological redshift derived according to the following three principles: the Stefan-Boltzmann law; the observed

T_{t} = T_{0} (1 + z)

relation; and the

R_{h} = c t

principle. We cannot see how the different Melia [12]

R_{h} = c t

cosmological model redshift formula can be consistent with all three principles simultaneously, but we are open to hearing other opinions on this topic. Despite this possible weakness in the Melia model, Melia [13,14] has done impressive and extensive research demonstrating how the

R_{h} = c t

principle appears to be favored by observations, when compared to the

Λ

-CDM model. We believe that different variants of

R_{h} = c t

cosmology models deserve increased attention and comparison in future studies.

References

E. G. Haug and E. T. Tatum. Solving the Hubble tension using the Union2 supernova database. Preprints.org, 2024a. [CrossRef]
E. G. Haug and E. T. Tatum. Solving the Hubble tension using the PantheonPlusSH0ES supernova database. Preprints.org, 2024b. [CrossRef]
E. G. Haug and E. T. Tatum. Planck length from cosmological redshifts solves the Hubble tension. ResearchGate.org, 2024c.
E. G. Haug. Closed form solution to the Hubble tension based on R_h=ct cosmology for generalized cosmological redshift scaling of the form: z=(R_h/R_t)^x-1 tested against the full distance ladder of observed SN Ia redshift. Preprints.org, 2024a. [CrossRef]
E. T. Tatum, U. V. S. Seshavatharam, and S. Lakshminarayana. The basics of flat space cosmology. International Journal of Astronomy and Astrophysics, 5:116, 2015. [CrossRef]
E. G. Haug and S. Wojnow. How to predict the temperature of the CMB directly using the Hubble parameter and the Planck scale using the Stefan-Boltzmann law. Journal of Applied Mathematics and Physics, 12:3552, 2024. [CrossRef]
E. G. Haug. CMB, Hawking, Planck, and Hubble scale relations consistent with recent quantization of general relativity theory. International Journal of Theoretical Physics, 63(57), 2024b. [CrossRef]
S. et. al Muller. A precise and accurate determination of the cosmic microwave background temperature at z=0.89. Astronomy & Astrophysics, 551, 2013. [CrossRef]
S. Dhal and R. K. Paul. Investigation on CMB monopole and dipole using blackbody radiation inversion. Scientific Reports, 13:3316, 2023. [CrossRef]
D.A. Riechers, A. Weiss, and F. et al. Walter. Microwave background temperature at a redshift of 6.34 from h₂o absorption. Nature, 602:58, 2022. [CrossRef]
L. Yunyang. Constraining cosmic microwave background temperature evolution with Sunyaev–Zel’dovich galaxy clusters from the atacama cosmology telescope. The Astrophysical Journal, 922:136, 2021. [CrossRef]
F. Melia. Cosmological redshift in Friedmann-Robertson-walker metrics with constant space-time curvature. Monthly Notices of the Royal Astronomical Society, 422:1418, 2012. [CrossRef]
F. Melia. Model selection with baryonic acoustic oscillations in the lyman-α forest. European Physics Letters, 143:59004, 2023. [CrossRef]
F. Melia. Strong observational support for the R_h=ct timeline in the early universe. Physics of the Dark Universe, 46:101587, 2024. [CrossRef]

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A Newly-Derived Cosmological Redshift Formula Which Solves the Hubble Tension and Yet Maintains Consistency with T_t = T_0(1 + z), the R_h = ct Principle and the Stefan-Boltzmann Law

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